Approximation of smooth surfaces with polygonal meshes
This is a very good question.
It all comes down to fooling your brain. 

1. “Approximating a curve with polygons is pretty crude”
Well, human eyes have considerably limited resolution.
Screen resolution also contributes - it isn't infinite.

You don't need infinite polygons, just enough to fool the human eye.
It's easier to do, than most people think.

2. “You would need billions of polygons for a smooth-looking surface”
Smooth-looking is the key word.

Surface needs far less geometry to look smooth, than a silhouette.
Most of your focus is on surfaces, not silhouettes.   

3. “I don't recall seeing a straight edge anywhere”
Don't recall is the key phrase.

You don't focus on details, but on the moving imagery.
When you pause the animation, chances are you'll find some jagged edges.  

Try pausing older movies like Toy story 1, Shrek or Ice age.
This is what Pixar understood first - you don't need it perfect. Just non-distracting.



Fig. 02 - Surface of a low-poly object looks perfectly smooth, while it's silhouette is jagged